📚 The Normal Distribution | 正态分布
The normal distribution is one of the most important continuous probability distributions in A-Level Mathematics. It is used to model a wide variety of natural phenomena, from heights and masses to measurement errors and exam scores. In Edexcel A-Level, you are expected to understand its properties, standardise using z-values, calculate probabilities using the standard normal table, and apply the normal distribution as an approximation to the binomial distribution under certain conditions.
正态分布是 A-Level 数学中最重要的连续型概率分布之一。它被用来模拟各种自然现象，从身高和体重到测量误差和考试成绩。在 Edexcel A-Level 考纲中，你需要理解它的性质，学会用 z 值进行标准化，使用标准正态分布表计算概率，并在特定条件下将正态分布作为二项分布的近似。
1. Understanding the Shape and Parameters | 理解曲线形状与参数
The normal distribution is defined by two parameters: the mean μ and the variance σ². The random variable is usually written as X ~ N(μ, σ²). The probability density function is bell-shaped and symmetric about the mean. The total area under the curve equals 1.
正态分布由两个参数定义：均值 μ 和方差 σ²。随机变量通常记作 X ~ N(μ, σ²)。概率密度函数呈钟形，并以均值为对称轴。曲线下的总面积等于 1。
The shape of the curve is determined by the standard deviation σ. A larger σ makes the curve flatter and more spread out, while a smaller σ yields a taller and narrower peak. Despite changes in μ and σ, the fundamental bell shape remains.
曲线的形状由标准差 σ 决定。σ 越大，曲线越扁平、散布越广；σ 越小，峰值越高、曲线越窄。尽管 μ 和 σ 发生变化，基本的钟形形状始终不变。
2. Properties of the Normal Curve | 正态曲线的性质
The normal curve has several key properties that are essential for solving problems. The mean, median and mode are all equal and located at the centre of the distribution. The curve is asymptotic to the horizontal axis, meaning it never actually touches the x-axis but extends infinitely in both directions.
正态曲线有几个关键的性质，这对于解题至关重要。均值、中位数和众数全部相等，并位于分布的中心。曲线以水平轴为渐近线，意味着它实际上永远不会触及 x 轴，但会向两端无限延伸。
Approximately 68% of the data lies within one standard deviation of the mean (μ ± σ), about 95% lies within two standard deviations (μ ± 2σ), and roughly 99.7% viations (μ ± 3σ). This is often referred to as the empirical rule or the 68-95-99.7 rule.
大约 68% 的数据落在均值的一个标准差范围内 (μ ± σ)，约 95% 落在两个标准差范围内 (μ ± 2σ)，大约 99.7% 落在三个标准差范围内 (μ ± 3σ)。这通常被称为经验法则或 68-95-99.7 法则。
- The curve is symmetric, so P(X < μ - k) = P(X > μ + k) for any k.
- 曲线是对称的，因此对任意 k，有 P(X < μ - k) = P(X > μ + k)。
3. The Standard Normal Distribution | 标准正态分布
To make probability calculations easier, any normal random variable X ~ N(μ, σ²) can be transformed into the standard normal variable Z ~ N(0, 1). The transformation is done by subtracting the mean and dividing by the standard deviation. The formula is:
为了让概率计算更简单，任何正态随机变量 X ~ N(μ, σ²) 都可以转换为标准正态变量 Z ~ N(0, 1)。转换的方法是减去均值再除以标准差。其公式为：
Z = (X - μ) / σ
) = P(Z < z). Remember that the table typically gives the area to the left of the z-value.
这被称为标准化。一旦得到 z 值，我们就可以使用标准正态分布表（考试中通常会提供）来查找对应的累积概率 Φ(z) = P(Z < z)。请记住，表格通常给出的是 z 值左侧的面积。
4. Using the Standard Normal Table | 使用标准正态分布表
symmetry property: Φ(-z) = 1 - Φ(z). This is because the total area under the curve is 1.
标准正态分布表提供非负 z 值（最高约 3.49）的 Φ(z) 值。对于负的 z 值，我们利用对称性：Φ(-z) = 1 - Φ(z)。这是因为曲线下的总面积为 1。
Suppose X ~ This is the probability. For the area to the right, use P(X > 115) = 1 - 0.8413 = 0.1587.
假设 X ~ N(100, 15²)，我们需要求 P(X < 115)。首先计算 z = (115 - 100)/15 = 1.00。然后查表得 Φ(1.00) ≈ 0.8413。这就是所求概率。对于右侧面积，使用 P(X > 115) = 1 - 0.8413 = 0.1587。
| z | 0.00 | 0.01 | 0.02 | 0.03 |
|---|---|---|---|---|
| 1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 |
5. Calculating Probabilities for Intervals | 计算区间概率
he standardised value of a and z₂ is the standardised value of b. Always draw a sketch to avoid sign errors.
要计算 X 落在两个值 a 和 b 之间的概率，我们使用 P(a < X < b) = Φ(z₂) - Φ(z₁)，其中 z₁ 是 a 标准化后的值，z₂ 是 b 标准化后的值。始终画一个草图以避免符号错误。
For example, if X ~ n probability = Φ(1.0) - Φ(-0.5) = Φ(1.0) - [1 - Φ(0.5)] = 0.8413 - (1 - 0.6915) = 0.5328.
例如，若 X ~ N(75, 10²)，求 P(70 < X < 85)。对两个界限标准化：z₁ = (70 - 75)/10 = -0.5，z₂ = (85 - 75)/10 = 1.0。那么概率为 Φ(1.0) - Φ(-0.5) = 0.8413 - (1 - 0.6915) = 0.5328。
6. Inverse Normal Calculations | 逆正态计算
Sometimes you are given a probability and need to find the corresponding value of X. This is known as the inverse normal problem. You first use the percentage points table (or the main table in reverse) to find the z-value that corresponds to the given cumulative probability.
有时题目给出一个概率，需要你找到对应的 X 值。这被称为逆正态问题。你首先使用百分点表（或反过来查主表）来找到与给定累积概率对应的 z 值。
Once you have the z-value, solve for X using the formula: X = μ + zσ. For instance, if X ~ N(50, 4²) and you want the value above which the top 5% h means P(Z < z) = 0.95. From tables, z ≈ 1.6449. Then X = 50 + 1.6449 × 4 = 56.58 approx.
一旦得到 z 值，使用公式 X = μ + zσ 求解 X。例如，若 X ~ N(50, 4²)，你想要找出数据中最高 5% 所对应的下界值，需要找到 z 使得 P(Z > z) = 0.05，这意味着 P(Z < z) = 0.95。查表得 z ≈ 1.6449。然后 X = 50 + 1.6449 × 4 = 56.58 左右。
Remember that Edexcel often provides a separate small table of common percentage points for 90%, 95%, 99% etc., which can save time.
请记住，Edexcel 通常会提供一个小型常用百分点表，如 90%、95%、99% 等，这可以节省时间。
7. Finding the Mean or Standard Deviation | 求解均值或标准差
Exam questions may require you to determine unknown μ or σ given certain probability statements. You set up equations using the standardisation formula and solve simultaneously if both are unknown, or solve directly if one parameter is missing.
考试题目可能会要求你在给定某些概率条件的情况下确定未知的 μ 或 σ。你需要利用标准化公式建立方程，如果两个参数都未知则联立求解，或者如果只有一个参数缺失则直接求解。
For example, given X ~ < 80)=0.9. Use z₁ ≈ -0.8416, z₂ ≈ 1.2816. Solve the two linear equations to find μ and σ.
例如，给定 X ~ N(μ, σ²)，P(X < 60) = 0.2 且 P(X > 80) = 0.1。将每个概率转化为 z 分数：(60 - μ)/σ = Φ⁻¹(0.2) 以及 (80 - μ)/σ = Φ⁻¹(0.9)，因为 P(X > 80)=0.1 意味着 P(X < 80)=0.9。使用 z₁ ≈ -0.8416，z₂ ≈ 1.2816。解这两个线性方程以求得 μ 和 σ。
Careful layout of the equations is essential to avoid algebraic mistakes.
谨慎地列出方程对于避免代数错误至关重要。
8. Normal Approximation to the Binomial | 用正态分布近似二项分布
When a binomial distribution X ~ 0.5 (or np and nq both greater than 5), it can be approximated by a normal distribution Y ~ imation simplifies calculations for cumulative probabilities that would otherwise be very tedious.
当二项分布 X ~ B(n, p) 的 n 很大且 p 接近 0.5（或 np 和 nq 均大于 5）时，可以用正态分布 Y ~ N(np, npq) 来近似，其中 q = 1 - p。这种近似简化了累积概率的计算，否则会非常繁琐。
Y < 55.5), and P(X ≥ 48) becomes P(Y > 47.5). For strict inequalities, adjust accordingly.
这个近似中关键的一步是连续性校正。因为二项分布是离散的而正态分布是连续的，我们将端点调整 0.5。例如，P(X ≤ 55) 变成 P(Y < 55.5)，P(X ≥ 48) 变成 P(Y > 47.5)。对于严格不等式，相应地进行调整。
| Discrete condition | Normal approximation with continuity correction |
|---|---|
| P(X < k) | P(Y < k - 0.5) |
| P(X ≤ k) | P(Y < k + 0.5) |
| P(X > k) | P(Y > k + 0.5) |
| P(X ≥ k) | P(Y > k - 0.5) |
Always state that you are using a continuity correction to earn full marks.
务必说明你正在使用连续性校正，以获得全部分数。
9. Sample Mean Distribution and Central Limit Theorem | 样本均值分布与中心极限定理
ample of size n is drawn from a population that is normally distributed, the sample mean X̄ , the Central Limit Theorem (CLT) states that for sufficiently large n (usually n ≥ 30), X̄ is approximately normal with the same parameters.
如果从正态分布的总体中抽取一个容量为 n 的随机样本，样本均值 X̄ 也服从正态分布，其均值为 μ，方差为 σ²/n。即使总体不是正态分布，中心极限定理 (CLT) 指出，对于足够大的 n（通常 n ≥ 30），X̄ 近似服从具有相同参数的正态分布。
When using the CLT, the standard deviation of the sample mean is the standard error: σ/√n. This concept is often tested in hypothesis testing questions in the applied statistics part of the A-Level.
当使用中心极限定理时，样本均值的标准差是标准误：σ/√n。这一概念经常在 A-Level 应用统计学的假设检验题目中被考查。
36 is taken. Find the probability that the sample mean is less than 195. The sample mean X̄ ~ N(200, 30²/36) = N(200, 5²). Then P(X̄ < 195) = Φ((195-200)/5) = Φ(-1.0) = 0.1587.
例如，某总体 μ = 200，σ = 30。抽取一个包含 36 个样本的样本组。求样本均值小于 195 的概率。样本均值 X̄ ~ N(200, 30²/36) = N(200, 5²)。那么 P(X̄ < 195) = Φ((195-200)/5) = Φ(-1.0) = 0.1587。
10. Solving Real-World Problems with the Normal Distribution | 用正态分布解决实际问题
A common style of Edexcel question gives a real-life context, such as the mass of loaves of bread or the length of bolts produced by a machine. You are often asked to find the proportion of items that meet a specification, or the probability that an item is rejected.
Edexcel 常见的题型是给出一个现实背景，例如面包的质量或机器生产的螺栓长度。题目经常要求你找出符合规格的产品比例，或者某个产品被拒绝的概率。
Another involved task is to determine the setting of a machine (changing the mean or standard deviation) to achieve a certain acceptance rate. This requires setting up the normal model and solving for the unknown parameter using the inverse normal.
另一种较复杂的任务是确定机器的设置（改变均值或标准差）以达到某个合格率。这需要建立正态模型，并使用逆正态求解未知参数。
Always define your random variable clearly at the start, such as 'Let X be the mass of a loaf in grams, X ~ N(400, 15²)'. Marks are awarded for accurate notation and interpretation of the results in context.
一定要在开始时清晰地定义随机变量，例如“设 X 为面包的质量（克），X ~ N(400, 15²)”。准确的符号表示和结合上下文对结果进行解释能够赚得分数。
11. Common Mistakes and How to Avoid Them | 常见错误及如何避免
A frequent error is forgetting to square the standard deviation when writing the distribution. N(μ, σ²) uses variance, so if the standard deviation is 5, the second parameter is 25. Always double-check your notation.
一个常见的错误是在书写分布时忘记将标准差平方。N(μ, σ²) 使用的是方差，因此如果标准差是 5，第二个参数应为 25。务必复查你的符号。
Another mistake is misreading the inequality direction when looking up tables. Sketch a bell curve and shade the required area before looking up z-values. Avoid subtracting from 1 incorrectly.
另一个错误是在查表时读反了不等号方向。在查找 z 值之前画一条钟形曲线并涂上所需的面积。避免错误地从 1 中减去。
When using continuity correction for the normal approximation, students often apply the wrong adjustment. Memorising the table of corrections or understanding the geometric reasoning behind the 0.5 shift will prevent errors.
在使用正态近似的连续性校正时，学生经常进行调整的方式错误。记住校正表格或理解 0.5 位移背后的几何推理可以防止出错。
Finally, avoid rounding z-values too early; keep at least three decimal places during intermediate steps to maintain accuracy in the final answer.
最后，避免过早舍入 z 值；在中间步骤中至少保留三位小数，以保证最终答案的精度。
12. Preparing for the Edexcel Exam | 备战 Edexcel 考试
In the Edexcel A-Level Mathematics exam, normal distribution questions appear in both the pure and statistics sections. They are often combined with binomial distribution, hypothesis testing, or calculations involving the sample mean. Expect a mix of straightforward probability computations and more demanding reverse problems.
在 Edexcel A-Level 数学考试中，正态分布题目会同时出现在纯数学和统计学部分。它们经常与二项分布、假设检验或涉及样本均值的计算结合起来。预计题目会混合直接的求概率计算和更有挑战性的逆向问题。
Practise using the formula booklet efficiently. The booklet provides the standard normal distribution table, the percentage points table, and the formula for standardising. Being familiar with the layout before the exam saves time and reduces stress.
高效地使用公式手册进行练习。手册提供了标准正态分布表、百分点表以及标准化公式。考前熟悉其排版可以节省时间并减轻压力。
When revising, work through past papers and focus on showing clear steps: state the distribution, standardise, find z, use table, and finally write the answer in context. Clear communication of your method is key to securing method marks.
复习时，通过历年真题进行练习，并注重展示清晰的步骤：阐明分布、标准化、求 z 值、查表，最后在上下文中写下答案。清晰地表达你的方法是获得方法分的关键。
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